Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8475 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,5,8,6,7,9,10)(11,12)(13,21)(14,25,27,18)(15,28)(16,20)(17,23)(19,31)(22,37,39,42)(24,35)(26,34,30,36)(29,33)(32,38,41,40);(4,7)(5,9)(11,13)(12,17)(14,22)(15,29)(16,24)(18,32)(19,35)(20,33)(21,28)(23,31)(25,41)(26,40)(27,39)(30,38)(34,37)(36,42);(4,7)(5,9)(11,14)(12,18)(13,22)(15,30)(16,27)(17,32)(19,36)(20,25)(21,37)(23,40)(24,39)(26,31)(28,34)(29,38)(33,41)(35,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94298);